Sure, with caveats. The ATF and the State of Texas say that private party sales are legal so long as you do not have knowledge that the person you are selling to is not a “prohibited person” who cannot legally own a firearm in Texas.
Admittedly, this is a low bar. However, it is wise to take prudent steps. Always draw up a bill of sale or contract stating what has been sold by who and to whom. Annotate what identification was provided, and have the purchaser sign stating that they are not a prohibited person according to the ATF definition.
An ounce of prevention goes a long way.
Alternatively, we are happy to sell your firearm for you, do the marketing, and most importantly perform a NICS check to ensure that the firearm is not falling into the wrong hands or exposing you to any liability. We charge 10% of the final sale price for this service and piece of mind.